Can we exponentiate d/dx? Vector (fields)? What is exp? | Lie groups, algebras, brackets #4

Can we exponentiate vectors? What does e^(d/dx) mean? Does it make sense to exponentiate a whole bunch of vectors? Well yes! While what these exponentials do seem very different at first, they can be recast into the same framework.

Files for download:

CHAPTERS:
00:00 Introduction
01:03 What is exponentiation?
04:15 Exponentiating vectors
11:23 Exponentiating derivatives
24:04 Exponentiating vector fields

❗Remark❗

1️⃣ I know that many people would be thinking of series expansion of exponentials. I deliberately avoid this because it is not conducive to learning the intuition of the exponential, and more crucially, it does not apply to the exponential of vectors on manifolds. The result is very manifold-dependent, and I will be very impressed if there is a series-like explanation for the exponential map in differential geometry.

2️⃣ However, I want to know: is there a generalisation of the translation operator statement in the video to manifolds? For a flat plane, we have exp(a * nabla) f(x) = f(x + a). And in fact,the exponential map on the flat manifold of R^n gives x + a = exp_x (a). Hence, for flat R^n, we have exp(a * nabla) f(x) = f(exp_x(a)). Can this be generalised to general manifolds? Is it true if I interpret nabla as not a normal gradient, but covariant derivative? Please let me know if you have any ideas for it. I want this to be true because it connects different “exponential” ideas.

📖 Further reading 📖

1️⃣ Exp vectors

❗You might also need to learn these first before tackling the link above❗

2️⃣ Exp d/dx

3️⃣ Exp vector field

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